Questions — CAIE P2 (699 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE P2 2016 November Q4
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + a x + 4$$ where \(a\) is a constant.
  1. Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { p } ( x )\) for all values of \(a\).
  2. Given that the remainder is - 42 when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ), find the value of \(a\).
  3. When \(a\) has the value found in part (ii), factorise \(\mathrm { p } \left( x ^ { 2 } \right)\) completely.
CAIE P2 2016 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-2_556_844_1731_648} The diagram shows the curve \(y = \frac { 4 \ln x } { x ^ { 2 } + 1 }\) and its stationary point \(M\). The \(x\)-coordinate of \(M\) is \(m\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m = \mathrm { e } ^ { 0.5 \left( 1 + m ^ { - 2 } \right) }\).
  2. Use an iterative formula based on the equation in part (i) to find the value of \(m\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2016 November Q6
6
  1. Show that \(\frac { \cos 2 \theta } { 1 + \cos 2 \theta } \equiv 1 - \frac { 1 } { 2 } \sec ^ { 2 } \theta\).
  2. Solve the equation \(\frac { \cos 2 \alpha } { 1 + \cos 2 \alpha } = 13 + 5 \tan \alpha\) for \(0 < \alpha < \pi\).
  3. Find the exact value of \(\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \cos x } { 1 + \cos x } \mathrm {~d} x\).
CAIE P2 2016 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-3_533_698_735_717} The diagram shows the curve with parametric equations $$x = 4 \sin \theta , \quad y = 1 + 3 \cos \left( \theta + \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant \theta < 2 \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(k ( 1 + ( \sqrt { } 3 ) \tan \theta )\) where the exact value of \(k\) is to be determined.
  2. Find the equation of the normal to the curve at the point where the curve crosses the positive \(y\)-axis. Give your answer in the form \(y = m x + c\), where the constants \(m\) and \(c\) are exact.
CAIE P2 2016 November Q1
1 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 4 } { x _ { n } ^ { 2 } } + \frac { 2 x _ { n } } { 3 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).
  1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  2. State an equation that is satisfied by \(\alpha\), and hence find the exact value of \(\alpha\).
CAIE P2 2016 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{8b051aee-4920-42a0-8b74-cbfa9f3c1ab1-2_429_821_826_662} The variables \(x\) and \(y\) satisfy the equation \(y = K x ^ { p }\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 1.28,3.69 )\) and \(( 2.11,4.81 )\), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.
  1. Find \(\int \tan ^ { 2 } 4 x \mathrm {~d} x\).
  2. Without using a calculator, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } ( 4 \cos 2 x + 6 \sin 3 x ) \mathrm { d } x\).
CAIE P2 2016 November Q4
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 3 x ^ { 2 } + 4 a x - 5 ,$$ where \(a\) is a constant. It is given that ( \(2 x - 1\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Use the factor theorem to find the value of \(a\).
  2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has only one real root.
  3. Use logarithms to solve the equation \(\mathrm { p } \left( 6 ^ { y } \right) = 0\) correct to 3 significant figures.
CAIE P2 2016 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{8b051aee-4920-42a0-8b74-cbfa9f3c1ab1-3_373_623_260_760} The diagram shows the curve \(y = \sqrt { } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 3 } x } \right)\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\).
  1. Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 2 decimal places.
  2. With reference to the diagram, explain why this estimate is greater than the exact area of \(R\).
  3. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.
CAIE P2 2016 November Q6
6 A curve has parametric equations $$x = \ln ( t + 1 ) , \quad y = t ^ { 2 } \ln t$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the exact value of \(t\) at the stationary point.
  3. Find the gradient of the curve at the point where it crosses the \(x\)-axis.
  4. Express \(\sin 2 \theta ( 3 \sec \theta + 4 \operatorname { cosec } \theta )\) in the form \(a \sin \theta + b \cos \theta\), where \(a\) and \(b\) are integers.
  5. Hence express \(\sin 2 \theta ( 3 \sec \theta + 4 \operatorname { cosec } \theta )\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  6. Using the result of part (ii), solve the equation \(\sin 2 \theta ( 3 \sec \theta + 4 \operatorname { cosec } \theta ) = 7\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2017 November Q1
1 Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} Write your Centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50. This document consists of 11 printed pages and 1 blank page. 1 Solve the equation \(\ln ( 3 x + 1 ) - \ln ( x + 2 ) = 1\), giving your answer in terms of e.
CAIE P2 2017 November Q2
2 Solve the equation \(5 \cos \theta ( 1 + \cos 2 \theta ) = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2017 November Q3
3 It is given that the variable \(x\) is such that $$1.3 ^ { 2 x } < 80 \quad \text { and } \quad | 3 x - 1 | > | 3 x - 10 | .$$ Find the set of possible values of \(x\), giving your answer in the form \(a < x < b\) where the constants \(a\) and \(b\) are correct to 3 significant figures.
CAIE P2 2017 November Q4
4
  1. Find \(\int \frac { 4 + \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } \mathrm {~d} \theta\).
  2. Given that \(\int _ { 0 } ^ { a } \frac { 2 } { 3 x + 1 } \mathrm {~d} x = \ln 16\), find the value of the positive constant \(a\).
CAIE P2 2017 November Q5
5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 37 x + 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\). It is also given that the remainder is 40 when \(\mathrm { p } ( x )\) is divided by ( \(2 x - 1\) ).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2017 November Q6
6 The parametric equations of a curve are $$x = 2 \mathrm { e } ^ { 2 t } + 4 \mathrm { e } ^ { t } , \quad y = 5 t \mathrm { e } ^ { 2 t }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and hence find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places.
  2. Find the gradient of the normal to the curve at the point where the curve crosses the \(x\)-axis.
CAIE P2 2017 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{ebfaafca-3279-44f8-a910-0756ffa25c70-10_542_789_260_676} The diagram shows the curve $$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$ The curve crosses the \(y\)-axis at the point \(P\) and the gradient of the curve at \(P\) is \(m\). The point \(Q\) on the curve has \(x\)-coordinate \(q\) and the gradient of the curve at \(Q\) is \(- m\).
  1. Find the value of \(m\) and hence show that \(q\) satisfies the equation $$x = a \sin \frac { 1 } { 2 } x + b ,$$ where the values of the constants \(a\) and \(b\) are to be determined.
  2. Show by calculation that \(- 4.5 < q < - 4.0\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2017 November Q1
1 Use logarithms to solve the equation \(5 ^ { 3 x - 1 } = 2 ^ { 4 x }\), giving your answer correct to 3 significant figures.
CAIE P2 2017 November Q2
2 It is given that \(x\) satisfies the equation \(| x + 1 | = 4\). Find the possible values of $$| x + 4 | - | x - 4 | .$$
CAIE P2 2017 November Q3
3 The equation of a curve is \(y = \tan \frac { 1 } { 2 } x + 3 \sin \frac { 1 } { 2 } x\). The curve has a stationary point \(M\) in the interval \(\pi < x < 2 \pi\). Find the coordinates of \(M\), giving each coordinate correct to 3 significant figures.
CAIE P2 2017 November Q4
4 The polynomials \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are defined by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } + a x - 15 \quad \text { and } \quad \mathrm { q } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + b x + 21 ,$$ where \(a\) and \(b\) are constants. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\) and also of \(\mathrm { q } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. Show that the equation \(\mathrm { q } ( x ) - \mathrm { p } ( x ) = 0\) has only one real root.
    \includegraphics[max width=\textwidth, alt={}, center]{e2b16207-2cb7-412b-ba7f-758e4d3f1ffb-06_631_643_260_749} The diagram shows the curve \(y = 4 e ^ { - 2 x }\) and a straight line. The curve crosses the \(y\)-axis at the point \(P\). The straight line crosses the \(y\)-axis at the point \(( 0,9 )\) and its gradient is equal to the gradient of the curve at \(P\). The straight line meets the curve at two points, one of which is \(Q\) as shown.
CAIE P2 2017 November Q6
6
  1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin x ( 4 \sin x + 6 \cos x ) \mathrm { d } x\).
  2. Given that \(\int _ { 0 } ^ { a } \frac { 6 } { 3 x + 2 } \mathrm {~d} x = \ln 49\), find the value of the positive constant \(a\).
CAIE P2 2017 November Q7
7 The equation of a curve is \(x ^ { 2 } + 4 x y + 2 y ^ { 2 } = 7\).
  1. Find the equation of the tangent to the curve at the point \(( - 1,3 )\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
  2. Show that there is no point on the curve at which the gradient is \(\frac { 1 } { 2 }\).
CAIE P2 2017 November Q1
1 Solve the equation \(\ln ( 3 x + 1 ) - \ln ( x + 2 ) = 1\), giving your answer in terms of e.
CAIE P2 2017 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{da5162f3-b5d5-417f-9b6c-5ae0024f22d9-10_542_789_260_676} The diagram shows the curve $$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$ The curve crosses the \(y\)-axis at the point \(P\) and the gradient of the curve at \(P\) is \(m\). The point \(Q\) on the curve has \(x\)-coordinate \(q\) and the gradient of the curve at \(Q\) is \(- m\).
  1. Find the value of \(m\) and hence show that \(q\) satisfies the equation $$x = a \sin \frac { 1 } { 2 } x + b ,$$ where the values of the constants \(a\) and \(b\) are to be determined.
  2. Show by calculation that \(- 4.5 < q < - 4.0\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2018 November Q1
1
  1. Solve the equation \(| 9 x - 2 | = | 3 x + 2 |\).
  2. Hence, using logarithms, solve the equation \(\left| 3 ^ { y + 2 } - 2 \right| = \left| 3 ^ { y + 1 } + 2 \right|\), giving your answer correct to 3 significant figures.